On Privacy Amplification, Lossy Compression, and Their Duality to Channel Coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. The converse significantly improves on previous bounds based on smooth min-entropy by Watanabe and Hayashi <xref ref-type="bibr" rid="ref7">[7]</xref> and turns out to be equivalent to a recent formulation in terms of the <inline-formula> <tex-math notation="LaTeX">$E_\gamma $ </tex-math></inline-formula> divergence by Yang <italic>et al.</italic> <xref ref-type="bibr" rid="ref9">[9]</xref>. In the latter, we show that the protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with the known relations between channel simulation and lossy source coding, this implies that the privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to protocols of the optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in <xref ref-type="bibr" rid="ref15">[15]</xref>, we show that the linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a “curious duality” in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia <xref ref-type="bibr" rid="ref16">[16]</xref> and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.